Theory of Unsteady Propeller Forces 



the pitch 27Th(r). Denoting the strength of the bound vortex in the elemental area 

 at point (r,v) at time t by 7^(r,v,t) dvdr, we can set 



7^(r,v,t) = 7j[r,v,t- 277(k- l)/(Nn)] , "(9) 



because the hydrodynamic state of flow field has the periodicity of period 27T/(Nn). 

 The doublets representing the blade thickness are distributed on Sp in the chord- 

 wise direction, and their strengths are approximately proportional to the product 

 of the blade thickness t (r,v) and the chordwise component of the mean velocity, 

 where the mean velocity is the velocity averaged over the chord of the blade 

 section. The similar procedure can be applied for the rudder. That is, the 

 bound vortex of the rudder is arranged approximately along the line u = constant 

 on the XjYj plane instead of the surface s^, and its strength in the elemental 

 area at point (yi>u) at time t is denoted by 7R(yi,u,t) dudy^. The free vortex 

 shed from it extends straight rearward in the Xjyj plane. Further the rudder 

 thickness can be represented by doublets distributed on the x^y^ plane in the x 

 direction, whose strengths are approximately proportional to the product of the 

 thickness tR(yj,u) and the mean velocity component along the chord. Thus, de- 

 noting the velocity potentials due to the propeller and the rudder by i>p and 0p 

 respectively, they can be obtained from the vortex systems and doublet distribu- 

 tions. By using subscripts I and t for the vortex systems (load) and the doublet 

 distributions (thickness) respectively, the total velocity potential s^ at point 

 (x,y,z) or (x, r,i9) at time t due to the ship hull, propeller, and rudder is 



= 0JJ + (/)p + c/)j^ , 0p = (/)pj + 0p^ , 0JJ = 0jj£ + 0jj^ . (10) 



Then denoting the velocity components induced by ^ in the x, y, z, r, and 8 

 directions by w^, Wy, w^, w^., and w^ respectively, we get 



'dip 'deb dd) 'dd> dd> /<i\ 



^x = ^' %=^' *z=-^' *r=^' ''0=TW' (11) 



and the following relations among them are obtained: 



Wj. = w COS + w^ sin 6 , w^ = -w sin (9 + w^ cos . (12) 



On the other hand we cannot neglect the effect of viscous velocity on the 

 flow field around the actual ship, where the viscous velocity, i.e., the velocity 

 caused by viscosity, is equal to the remainder obtained by subtracting the veloc- 

 ity induced by the velocity potential from the actual velocity; the viscous veloc- 

 ity appears in the boundary layer and wake of the hull. We denote the components 

 of viscous velocity in the x, y, z, r, and 9 directions by v^^, Vjy, Vj^, v^^, 

 and Vjg respectively and get the following relations among them: 



Vjr = Vjy cos e + Vj^ sin 6 , v^^ = -Vjy sin + Vj^ cos 9 . (13) 



Further the condition of continuity must be satisfied as follows: 



9x By Bz Bx Br r rB6 



21 



. (14) 



